1.2 Financial Planning and Control 23
∑
s
H
···
∑
s
1
p(s
1
,...,s
H
)(−rw(s
1
,...,s
H
)+qy(s
1
,...,s
H
)) .
The first-period constraint is simply to invest the initial wealth:
∑
i
x(i,1)=b .
The constraints for periods t = 2,...,H are, for each s
1
,...,s
t−1
:
∑
i
−
ξ
(i,t −1,s
1
,...,s
t−1
)x(i,t −1, s
1
,...,s
t−2
)
+
∑
i
x(i,t,s
1
,...,s
t−1
)=0 ,
while the constraints for period H are:
∑
i
ξ
(i,H,s
1
,...,s
H
)x(i,H, s
1
,...,s
H−1
)
−y(s
1
,...,s
H
)+w(s
1
,...,s
H
)=G .
Other constraints restrict the variables to be non-negative.
To specify the model in this example, we use initial wealth, b = 55,000 ; target
value, G = 80,000 ; surplus reward, q = 1 ; and shortage penalty, r = 4.There-
sult is a stochastic program in the following form where the units are thousands of
dollars:
maxz =
2
∑
s
1
=1
2
∑
s
2
=1
2
∑
s
3
=1
0.125(y(s
1
,s
2
,s
3
) −4w(s
1
,s
2
,s
3
)) (2.1)
s. t. x(1,1)+x(2,1)=55 ,
−1.25x(1,1) −1.14x(2, 1)+x(1,2,1)+x(2,2,1)=0 ,
−1.06x(1,1) −1.12x(2, 1)+x(1,2,2)+x(2,2,2)=0 ,
−1.25x
(1,2, 1) −1.14x(2,2, 1)+x(1,3,1,1)+x(2,3,1,1)=0 ,
−1.06x(1,2,1) −1.12x(2,2, 1)+x(1,3,1,2)+x(2, 3,1,2)=0 ,
−1.25x(1,2,2) −1
.14x(2,2, 2)+x(1,3,2,1)+x(2,3, 2,1)=0 ,
−1.06x(1,2,2) −1.12x(2,2, 2)+x(1,3,2,2)+x(2, 3,2,2)=0 ,
1.25x(1,3,1, 1)+1.14x(2,3, 1
,1) −y(1,1, 1)+w(1,1,1)=80 ,
1.06x(1,3,1, 1)+1.12x(2,3, 1,1) −y(1,1, 2)+w(1,1,2)=80 ,
1.25x(1,3,1, 2)+1.14x(2,3, 1,2) −y(1,2, 1
)+w(1,2,1)=80 ,
1.06x(1,3,1, 2)+1.12x(2,3, 1,2) −y(1,2, 2)+w(1,2,2)=80 ,
1.25x(1,3,2, 1)+1.14x(2,3, 2,1) −y(2,1, 1)+w(2,1,1)=80 ,
1.06x(1,3,2, 1)+1.12x(2,3, 2,1) −y(2,1, 2)+w(2,1,2)=80 ,
1.25x(1,3,2, 2)+1.14x(2,3, 2,2) −y(2,2, 1)+w(2,2,1)=80 ,
1.06x(1,3,2,
2)+1.12x(2, 3,2,2) −y(2,2,2)+w(2,2, 2)=80 ,
x(i,t,s
1
,...,s
t−1
) ≥0 , y(s
1
,s
2
,s
3
) ≥ 0 , w(s
1
,s
2
,s
3
) ≥ 0 ,
for all i,t,s
1
,s
2
,s
3
.